Problem 11 - Algebra 1: Concepts and Skills

Question

Classify the model as exponential growth or exponential decay. $$ y=55(0.3)^{t} $$

Step-by-Step Solution

Answer

The model is classified as exponential decay.

Step 1: Identify the function type

Identify the form of a function in the form \(y = ab^{t}\). In this case, a = 55 and b = 0.3.

Step 2: Classify the exponential function

Classify whether it's exponential growth or decay by checking the value of b. If b > 1, it's exponential growth. If 0 < b < 1, it's exponential decay. Since b = 0.3, which is less than 1 and greater than 0, it can be said that it's an exponential decay.

Key Concepts

Exponential Functions

Exponential functions are an essential part of mathematics, helping us understand how quantities change rapidly in both nature and human applications. In general, an exponential function can be described as follows:

• It's a mathematical expression in the form of \( y = ab^t \), where \( a \) is a constant, \( b \) is the base, and \( t \) is the exponent or time.
• The base \( b \) is crucial as it determines the nature of growth or decay - whether the value increases or decreases as time progresses.
• If \( a = 1 \) and \( b = 2 \), over each unit time period, the value of \( y \) doubles.

Exponential functions are used in various fields such as finance, science, and engineering. They model situations where change happens at a rate proportional to the current value, making them indispensable for representing real-world problems. Understanding their characteristics allows us to predict or analyze patterns effectively.

Algebra

Algebra is the branch of mathematics that uses symbols and letters to represent numbers and quantities in mathematical expressions and equations. It is fundamental for solving problems involving exponential functions. Here's how you can use algebra when working with exponential functions:

• Start by identifying components of the function, such as the constants and variables.
• Manipulate the function using algebraic operations to solve for unknowns, such as isolating \( t \) or \( y \).
• Graphing is also an algebraic tool used frequently with exponential functions to visualize growth or decay over time.
• Recognize patterns such as consistent multipliers in exponential patterns, which helps determine growth or decay.

Incorporating algebraic techniques provides flexibility and precision in handling exponential relations. This skill is essential, not only for academic problems but also for practical scenarios where projections and estimations are needed.

Exponential Growth and Decay

Exponential growth and decay refer to how the quantity of something increases or decreases over time in a way that the rate of change is proportional to its current value. Let's dive into what differentiates these two:

• **Exponential Growth**: Occurs when the base \( b \) of the exponential function \( y = ab^t \) is greater than 1. In such cases, as time increases, the function value rapidly increases, often seen in populations, investments, and technologies.
• **Exponential Decay**: Happens when the base \( b \) is between 0 and 1. The function value decreases over time, a common pattern in processes such as radioactive decay or depreciation of assets.

The given problem \( y=55(0.3)^t \) is a perfect example of exponential decay:- The base (0.3) is less than 1, indicating that as time progresses, the value of \( y \) shrinks.Understanding these concepts clarifies how different phenomena behave over time and allows us to use mathematical models to predict future outcomes.